A method for predicting the fracture toughness of polyurethane grouting materials

Abstract

This invention belongs to the field of fracture toughness model prediction technology, and provides a method for predicting the fracture toughness of polyurethane grouting materials. Starting from the geometry of polyurethane, this invention considers the joint stress on the pore edges and pore surfaces under load, and establishes a calculation model for the fracture toughness of polyurethane materials based on density, temperature, and geometric characteristic values; it predicts the fracture toughness of polyurethanes with different densities at different temperatures. The prediction model of this invention can calculate the fracture toughness values ​​of polyurethane materials under different temperature environments based on the relative density and temperature of the material. The calculated values ​​agree well with the experimental values, providing a basis for the widespread application of polyurethane grouting materials.

Description

Technical Field

[0001] This invention relates to the field of fracture toughness model prediction technology, and in particular to a method for predicting the fracture toughness of polyurethane grouting materials. Background Technology

[0002] When polyurethane materials are used as structural materials in engineering, cracks easily form inside or on the surface of the material under load. The initiation and propagation of cracks are closely related to the material's repair effectiveness; therefore, studying the fracture properties of polymers is crucial. Compared to metals and concrete, the concept of flexural fracture is still in its early stages in polymers, but significant progress has been made in tensile fracture research. Most plastic foam materials gradually crumble under compression until they reach complete densification, and fail under tension through the propagation of individual cracks. Since cracks weaken the load-bearing capacity of the foam structure, fracture toughness has become an important characteristic for studying porous materials.

[0003] Closed-cell foams have a relatively complex structure. When liquid components are used to prepare foam, surface tension pulls the material towards the pore walls, leaving only a thin, easily ruptured film through the pores. Therefore, although closed-cell foams have a closed pore structure, their stiffness comes entirely from the walls surrounding the pores, and their modulus is equivalent to that of open-cell foams. However, some foams differ from this. Some polymer and glass-based foams have pores composed of truly solid parts, which increases the rigidity of the porous material. For example, many natural porous materials (such as leaves) have closed pores with thick pore surfaces.

[0004] Current research on the fracture properties of polyurethane grouting materials is somewhat insufficient. Firstly, studies on the fracture performance of polyurethane materials in extreme service environments are not comprehensive enough: existing studies mostly consider the flexural strength of polyurethane grouting materials, while research on the fracture process and other fracture parameters is lacking. Secondly, theoretical calculations of polyurethane materials lack comprehensive exploration. Many scholars have studied theoretical formulas for polymer fracture toughness, but most existing formulas neglect the influence of pore surface extension on fracture toughness when closed-cell foam fails.

[0005] Therefore, the development of a method to predict the fracture toughness of polyurethane grouting materials at different temperatures based on the material's relative density and temperature is of great significance. Summary of the Invention

[0006] The purpose of this invention is to provide a method for predicting the fracture toughness of polyurethane grouting materials in order to overcome the shortcomings of the prior art.

[0007] To achieve the above-mentioned objectives, the present invention provides the following technical solution:

[0008] This invention provides a method for predicting the fracture toughness of polyurethane grouting materials. The formula set for predicting the fracture toughness of polyurethane grouting materials at different temperatures is as follows:

[0009]

[0010] Among them, K ⅠC Fracture toughness of polyurethane grouting material, MPa·m 0.5 ρ*: Density of polyurethane grouting material, g / cm³ 3 ;ρ s Density of polyurethane material before foaming, g / cm³ 3 ;σ ys : Tensile strength of polyurethane grouting material at temperature T, MPa; d: Cell diameter, mm; T: Service temperature of polyurethane grouting material, °C.

[0011] As a preferred option, the polyurethane grouting material is a closed-cell material with a cell morphology of cubic prisms.

[0012] Preferably, when polyurethane grouting material is subjected to load, both the pore edges and the pore surface are stressed.

[0013] Preferably, the polyurethane grouting material has spherical pores and a pore wall thickness of 0.95–1.05 μm, and the pore wall thickness does not change with the density of the polyurethane grouting material.

[0014] Preferably, temperature affects the strength of polyurethane grouting materials by influencing molecular groups and molecular chains.

[0015] As a preferred option, σ ys =(-2.29+18.43ρ) * (1-0.0028T) is obtained by fitting the tensile strength of polyurethane grouting materials of different densities at different temperatures.

[0016] Preferably, 0.0353 and 0.284 are obtained by fitting the relative density and fracture toughness of the polyurethane grouting material at -40℃.

[0017] Preferably, the polyurethane grouting material is prepared from isocyanate and polyol.

[0018] The beneficial effects of this invention include:

[0019] 1) Starting from the geometry of polyurethane, this invention considers the stress on the pore edges and pore surfaces under load, and establishes a fracture toughness prediction formula and model for polyurethane grouting materials, providing a basis for the promotion and application of polyurethane grouting materials.

[0020] 2) Based on linear elastic fracture mechanics, and considering the closed-cell structure of polyurethane grouting materials from a microscopic perspective, a calculation model for the fracture toughness of polyurethane materials regarding density, temperature, and geometric characteristics was established using the parameter φ (volume fraction of solids contained in the edges of the polymer) proposed by Gibson-Ashby. The fracture toughness of polyurethanes with different densities at different temperatures was predicted. The prediction model of this invention can calculate the fracture toughness values ​​of polyurethane materials under different temperature environments based on the relative density and temperature of the material, and the calculated values ​​agree well with the experimental values.

[0021] 3) This invention estimates and corrects relevant parameters of the model using experimental data, and finally performs predictive analysis on the model. Comparing the theoretical and experimental data obtained from the predictive model of this invention with experimental data from relevant literature, it was found that the error between the experimental and theoretical values ​​of this invention is within 12%, while in relevant literature, the error between the experimental and calculated values ​​in 83.3% of cases is less than 10%. Attached Figure Description

[0022] Figure 1 A diagram of the closed-cell hexahedral structure of polyurethane grouting material;

[0023] Figure 2 This is a diagram showing the cell deformation of closed-cell foam under tensile stress.

[0024] Figure 3 Temperature-tensile strength fitting curves for polyurethanes of different densities;

[0025] Figure 4 The relative density and K of polyurethane grouting material at -40℃ IC / σ value fitting curve;

[0026] Figure 5 A comparison chart of fracture toughness test data and predicted theoretical data of polyurethane grouting material at 0℃.

[0027] Figure 6 A comparison chart of the fracture toughness test data and the predicted theoretical data of polyurethane grouting material at 20℃.

[0028] Figure 7 This is a comparison chart of the fracture toughness test data and the predicted theoretical data of polyurethane grouting material at 50℃. Detailed Implementation

[0029] The most important characteristic of porous solid materials compared to other materials is their relative density (the ratio of the density of the porous material to the density of the solid material that makes up the pore walls and pore surfaces), which has the most significant impact on the performance of foam materials.

[0030] When closed-cell materials are subjected to load, not only are the pore edges stressed, but the pore surfaces also break under membrane stress. The pore edges can bend, extend, or shrink, and the membrane on the pore surface will stretch, increasing the rigidity of the pore walls. The foam cells of polyurethane grouting materials have a closed-cell hexahedral structure, and the cell structure is as follows... Figure 1 As shown, let the side length of the cell body be l, and the thickness of the cell pore edge be t. e The hole surface thickness is t f The cell body has a square cross-section with a side length of l.

[0031] Plastic collapse causes the membrane of closed-cell foam materials to wrinkle along the direction perpendicular to the stretching. Because the membrane is relatively thin, the force required to break it is small. However, the membrane at the perpendicular direction will stretch, and the force required for membrane expansion has a significant impact on the properties of the foam material. When closed-cell foam materials are loaded, the bending of the pore edges causes the pore surface to stretch. The stretching direction under compressive loading is perpendicular to the stretching direction under tensile loading, but the magnitudes are similar in both cases. The deformation mechanism of the pores under tensile force is as follows: Figure 2 As shown, force F requires work to counteract the bending of the cell pore edges and the recovery of the pore surface extension.

[0032] The relationship between the fracture toughness of closed-cell foam materials and the thickness of the cell ridge and the cell surface is shown in Equation 1.

[0033]

[0034] In Equation 1, K ⅠC Fracture toughness of polyurethane grouting material, MPa·m 0.5 ;σ ys : Yield strength, MPa; l: Cell length, mm; t e : Thickness of the hole edge, mm; t f : Thickness of the hole surface, mm; α, β: coefficients to be determined.

[0035] Polyurethane grouting material is a closed-cell material, and the cell morphology is cubic prism.

[0036] and The calculation formulas are Equation 2 and Equation 3, respectively.

[0037] Materials

[0038]

[0039] In equations 2 and 3, t e Cell edge thickness, mm; t f : Cell surface thickness, mm; l: Cell length, mm; ρ*: Density of polyurethane grouting material, g / cm³ 3 ;ρ sDensity of the polyurethane material before foaming (the polyurethane material before molding without pores), in g / cm³ 3 φ: The volume fraction of solid material contained in the cavity edge of the polyurethane.

[0040] Combining equations 1 to 3, the relationship between polyurethane grouting material and cell parameters is shown in equation 4.

[0041]

[0042] In Equation 4, K ⅠC Fracture toughness of polyurethane grouting material, MPa·m 0.5 ρ*: Density of polyurethane grouting material, g / cm³ 3 ;ρ s Density of polyurethane material before foaming, g / cm³ 3 ; Φ: Volume fraction of cavity edge; l: Cell length, mm; σ ys : Yield strength, MPa; C5, C6: coefficients to be determined. The relationship between Φ and porosity and cell wall thickness is shown in Equation 5.

[0043]

[0044] In Equation 5, k: proportionality coefficient, which is 0.5; S v : Specific surface area of ​​the foam cells; δ: Foam cell wall thickness, mm; f: Porosity of the foam material.

[0045] Assuming the bubble pores are spherical and the specific surface area S v It is represented by Equation 6.

[0046]

[0047] In Formula 6, d: the average diameter of the cell body pores, mm; R: the average radius of the cell body pores, mm.

[0048] The cell wall thickness of the polyurethane grouting material is 1 μm, and it does not change with the density of the material, so it can be ignored. Equation 5 can be simplified to Equation 7.

[0049]

[0050] Porosity f refers to the percentage of gas in the total volume of the plastic foam. It is mainly affected by the density of the foam material itself and the density of the polyurethane material before foaming. The relationship between porosity and the relative density of the foam material is shown in Equation 8.

[0051]

[0052] As can be seen from the above equations, the relationship between the fracture toughness of polyurethane foam and the cell structure performance is shown in Equation 9.

[0053]

[0054] In Equation 9, K ⅠC Fracture toughness of polyurethane grouting material, MPa·m 0.5 ρ*: Density of polyurethane grouting material, g / cm³ 3 ;ρ s Density of polyurethane material before foaming, g / cm³ 3 ; d: average diameter of cell pores, mm; σ ys : Yield strength, MPa; C5, C6: Undetermined coefficients.

[0055] The inherent structure and physical properties of a material determine its performance variations. Polyurethane cell morphology varies significantly with different densities. Higher density polyurethane grouting materials exhibit more dispersed cell distribution and a denser matrix. The fracture toughness of the cell is directly proportional to its relative density. Therefore, it can be inferred that the difference in fracture toughness of polyurethane grouting materials under the same service environment is due to differences in cell morphology. At low densities, polyurethane cells are polyhedral with small contact surfaces between adjacent cells; higher-density polyurethane cells have isolated spherical structures. As density increases, the internal pore structure of the polyurethane grouting material becomes smaller, the elastic modulus increases, deformation decreases, and the material's fracture toughness increases. Furthermore, high-density polyurethane has a larger elastic modulus, and its ultimate load-bearing capacity is greater than that of low-density materials; therefore, its peak load at fracture increases with increasing density.

[0056] Temperature has little effect on the cell morphology of polyurethane foam, but mainly affects the strength properties of the material by influencing molecular groups and chains. The strength is related to parameters such as the yield strength and relative density of the polyurethane material before foaming. Gibson-Ashby simplified the complex theoretical model into a relationship between cell strength and temperature, as shown in Equation 10.

[0057]

[0058] In Equation 10, σ s : Yield strength of polyurethane grouting material at temperature T, MPa; σ s 0 Yield strength of polyurethane grouting material at absolute zero, MPa; T g Absolute zero, -273.15℃ (0K); β m : Constant coefficient (obtained from experimental data).

[0059] Based on the Gibson-Ashby formula, we propose to use formula σ sThe formula = f(ρ)(1-bT) is used to solve for the tensile strength of polyurethane materials at different temperatures. Here, f(ρ) is a parameter related to the material density, and b is set as a constant parameter. The temperature-tensile strength fitting curves for polyurethanes with different densities are shown below. Figure 3 As shown in Table 1, the temperature-tensile strength fitting parameters for polyurethanes of different densities are as follows.

[0060] Table 1. Temperature-tensile strength fitting parameters for polyurethanes of different densities.

[0061] <![CDATA[Density / (g / cm 3 )]]> f(ρ) b <![CDATA[R 2 ]]> 0.20 1.784 0.0028 0.869 0.25 2.164 0.0028 0.947 0.30 2.807 0.0028 0.935 0.35 3.913 0.0028 0.881 0.40 5.516 0.0028 0.895

[0062] By fitting the known f(ρ) value, the relationship between f(ρ) and the density ρ* of the polyurethane grouting material is obtained as shown in Equation 11.

[0063] f(ρ) = -2.29 + 18.43ρ *

[0064] R 2 =0.934 Equation 11

[0065] Therefore, the relationship between the tensile strength of polyurethane materials of different densities and temperature is shown in Equation 12.

[0066] σ ys =(-2.29+18.43ρ) * )(1-0.0028T) Formula 12

[0067] In Equation 12, σ ys ρ*: Tensile strength of polyurethane grouting material at temperature T, MPa; ρ*: Density of polyurethane grouting material, g / cm³ 3 T: Service temperature of polyurethane grouting material, °C.

[0068] The polyurethane grout is formed from isocyanate and polyol, and the relationship between the polyurethane pore diameter and the material density is shown in Equation 13.

[0069] d = -371.19ρ * +323.29 Equation 13

[0070] In Equation 13, d: average diameter of cell pores, mm; ρ*: density of polyurethane grouting material, g / cm³ 3 .

[0071] The relative density and K of polyurethane grouting material at -40℃ IC The values ​​are fitted, and the fitted curve is as follows: Figure 4 As shown in Table 2, the theoretical formula and empirical parameter values ​​are obtained, yielding the undetermined coefficients C5 and C6.

[0072] Table 2. Empirical Parameter Values ​​for Theoretical Formulas

[0073] <![CDATA[C5]]> <![CDATA[C6]]> <![CDATA[R 2 ]]> 0.0353 0.284 0.997

[0074] Fitted R 2 The value of 0.997 proves that the regression model can accurately reflect the changes in the data, and also indicates that the experimental data is relatively reliable.

[0075] In this invention, the theoretical formula set for predicting the fracture toughness of polyurethane materials at different temperatures is as follows:

[0076]

[0077] Among them, K ⅠC Fracture toughness of polyurethane grouting material, MPa·m 0.5 ρ*: Density of polyurethane grouting material, g / cm³ 3 ;ρ s Density of polyurethane material before foaming, g / cm³ 3 ;σ ys : Tensile strength of polyurethane grouting material at temperature T, MPa; d: Cell diameter, mm; T: Service temperature of polyurethane grouting material, °C.

[0078] The technical solutions provided by the present invention will be described in detail below with reference to the embodiments, but they should not be construed as limiting the scope of protection of the present invention.

[0079] The polyurethane grouting material of this invention is a two-component foamed polyurethane grouting material produced by Zhengzhou Anyuan Engineering Technology Co., Ltd. The raw materials of the two-component foamed polyurethane grouting material include component A and component B. Component A, model 9802A, is mainly isocyanate (WANNATE@CW20), and component B, model 9802B, is mainly polyol (WANOL@R2305), amine catalyst, and physical foaming agent. Components A and B undergo a polymerization reaction via a high-pressure gas atomization method to obtain the polyurethane grouting material. The density of the polyurethane grouting material is 0.20–0.40 g / cm³. 3 .

[0080] Example 1

[0081] To investigate the effects of temperature environment on dumbbell-shaped and cuboid polyurethane grouting materials (154mm x 17.5mm x 35mm), a high-low temperature alternating damp heat test chamber was used to simulate the service environment of the materials. A TTHH40W-EX temperature and humidity recorder was used to record the temperature and humidity changes inside the test chamber. The density of the polyurethane grouting materials was 0.20 g / cm³. 3 0.23g / cm 3 0.28g / cm 3 0.32g / cm 30.33g / cm 3 0.35g / cm 3 The corresponding relative densities were 0.167, 0.208, 0.250, 0.292, 0.308, and 0.333, respectively. The fracture toughness of the polyurethane grouting material was tested at 0℃.

[0082] Three-point bending fracture tests were conducted on polyurethane according to the standard test method for plane strain fracture toughness and strain energy release rate of plastic materials (ASTM D5045-1999(2007)e1). The specimen dimensions were: height W = 35 mm, span S = 140 mm, length L = 154 mm, width B = 17.5 mm; seam height ratio a / W = 0.4, initial seam height a = 17.5 mm. The three-point bending fracture test lasted for 7 days, with a seam height ratio of 0.4, a span of 140 mm, and a height of 35 mm.

[0083] The fracture toughness test data and theoretical data of the prediction model for polyurethane grouting materials at 0℃ are shown in Table 3. Figure 5 As shown.

[0084] Table 3. Experimental and theoretical data at 0℃

[0085]

[0086] Example 2

[0087] The temperature in Example 1 was changed from 0℃ to 20℃, and the density of the polyurethane grouting material was 0.20 g / cm³. 3 0.27g / cm 3 0.33g / cm 3 0.34g / cm 3 0.38g / cm 3 Under the same conditions as in Example 1, the fracture toughness test data and theoretical data of the polyurethane grouting material at 20℃ are shown in Table 4 and . Figure 6 As shown.

[0088] Table 4. Experimental and theoretical data at a temperature of 20℃

[0089]

[0090] Example 3

[0091] The temperature in Example 1 was changed from 0℃ to 50℃, and the density of the polyurethane grouting material was 0.20 g / cm³. 3 0.25g / cm 3 0.30g / cm 3 0.35g / cm 3 0.37g / cm3 0.40g / cm 3 Under the same conditions as in Example 1, the fracture toughness test data and theoretical data of the polyurethane grouting material at 50℃ are shown in Table 5 and . Figure 7 As shown.

[0092] Table 5. Experimental and theoretical data at a temperature of 50℃

[0093]

[0094] Example 4

[0095] We selected some data from literature related to the fracture toughness of closed-cell foam materials for statistical analysis and compared them with the calculated values ​​from the model in this paper. The results are shown in Table 6.

[0096] Table 6. Experimental data from different literature and theoretical data from this paper.

[0097]

[0098]

[0099] As can be seen from Examples 1 to 3, when the fracture toughness values ​​of polyurethane grouting materials measured under different test conditions are compared with the values ​​calculated by the theoretical model, the relative errors between the two are within 12%. The theoretical model can basically effectively and correctly predict the fracture toughness values ​​of polyurethane with different densities.

[0100] As shown in Example 4, 83.3% of the experimental data and calculated data in the relevant literature have a relative error of less than 10%, which shows that the prediction model has good applicability for predicting the fracture toughness of closed-cell foam materials.

[0101] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for predicting the fracture toughness of a polyurethane grouting material, characterized by, The formula group for predicting the fracture toughness of polyurethane grouting material at different temperatures is: wherein K ⅠC : fracture toughness of the polyurethane grouting material, MPa.m 0.5 ; p*: density of the polyurethane grouting material, g / cm 3 ; p s : density of the polyurethane material before foaming, g / cm 3 ; s ys : tensile strength of the polyurethane grouting material at temperature T, MPa; d: cell diameter, mm; T: service temperature of the polyurethane grouting material, °C; T ranges from -40 to 50°C, p s ranges from 0.20 to 0.40 g / cm 3 .

2. The prediction method of claim 1, wherein, The polyurethane grouting material is a closed-cell material, and the cell morphology is a cubic prism.

3. The prediction method according to claim 1 or 2, characterized in that, When the polyurethane grouting material is subjected to load, both the cell edges and the cell faces are stressed.

4. The prediction method of claim 3, wherein, The cell shape of the polyurethane grouting material is spherical, and the cell wall thickness is 0.95-1.05 μm, which does not change with the density of the polyurethane grouting material.

5. The prediction method of claim 3, wherein, Temperature affects the strength of the polyurethane grouting material by affecting the molecular groups and molecular chains.

6. The prediction method according to claim 5, characterized in that, σ ys = (-2.29 + 18.43p) (1 - 0.0028T) was fitted from tensile strengths of different density polyurethane grouts at different temperatures.

7. The prediction method of claim 4 or 6, characterized in that, 0.0353 and 0.284 are obtained by fitting the relative density and fracture toughness of the polyurethane grouting material at -40℃.

8. The prediction method of claim 7, wherein, The polyurethane grouting material is prepared from isocyanate and polyol.

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